AP Precalculus Practice Test: Unit 3 Question #8 Finding cos(5pi/6) Using the Unit Circle

Please subscribe!    / nickperich   My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May. *AP Precalculus Practice Test: Unit 3 Question #8* *Finding \(\cos\left(\frac{5\pi}{6}\right)\) Using the Unit Circle* --- *Key Concepts and Vocabulary* 1. **Unit Circle**: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The cosine of an angle is the \(x\)-coordinate of the point on the unit circle corresponding to that angle. 2. **Cosine Definition**: The cosine of an angle \(\theta\) in the unit circle is the horizontal coordinate of the corresponding point, or: \[ \cos(\theta) = x \] where \(x\) is the \(x\)-coordinate of the point on the unit circle at that angle. 3. **Reference Angle**: For angles in the second quadrant (like \(\frac{5\pi}{6}\)), the reference angle is the angle formed with the \(x\)-axis, which is found by subtracting the angle from \(\pi\). --- *Question Setup* We are tasked with finding \(\cos\left(\frac{5\pi}{6}\right)\) using the unit circle. --- *Step-by-Step Solution* 1. **Locate \(\frac{5\pi}{6}\) on the Unit Circle**: The angle \(\frac{5\pi}{6}\) lies in the second quadrant. Since the unit circle is symmetric, we know that \(\frac{5\pi}{6}\) is \(\frac{\pi}{6}\) away from \(\pi\). 2. **Reference Angle**: The reference angle for \(\frac{5\pi}{6}\) is: \[ \text{Reference Angle} = \pi - \frac{5\pi}{6} = \frac{\pi}{6} \] 3. **Use the Known Cosine Value for the Reference Angle**: From the unit circle, we know that: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \] Since the cosine value is negative in the second quadrant, we take the negative of this value. 4. **Final Answer**: Therefore, \[ \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \] --- *Final Answer* \[ \boxed{-\frac{\sqrt{3}}{2}} \] --- *Purpose of the Question* This problem assesses your ability to: Use the unit circle to find the cosine of an angle. Recognize the reference angle and apply the correct sign for the cosine in different quadrants. Recall key values for standard angles on the unit circle, such as \(\frac{\pi}{6}\), and use them to find cosine. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study